A note on the relationship between the Szlenk and $w^*$-dentability indices of arbitrary $w^*$-compact sets

TL;DR

This paper establishes the optimal relationship between the Szlenk and $w^*$-dentability indices of $w^*$-compact sets in dual Banach spaces, introducing a game-theoretic approach to analyze these indices.

Contribution

It provides the first optimal estimate linking Szlenk and $w^*$-dentability indices and introduces a two-player game to determine the Szlenk index of $w^*$-compact convex sets.

Findings

Established the optimal estimate between Szlenk and $w^*$-dentability indices.

Introduced a two-player game characterizing the Szlenk index of $w^*$-compact convex sets.

Applied results to analyze the $w^*$-dentability index of Banach spaces and operators.

Abstract

We prove the optimal estimate between the Szlenk and $w^*$-dentability indices of an arbitrary $w^*$-compact subset of the dual of a Banach space. For a given $w^*$-compact, convex subset $K$ of the dual of a Banach space, we introduce a two player game the winning strategies of which determine the Szlenk index of $K$. We give applications to the $w^*$-dentability index of a Banach space and of an operator.